SMC with adaptive resampling: Large sample asymptotics

A longstanding problem in sequential Monte Carlo (SMC) is to mathematically prove the popular belief that resampling does improve the performance of the estimation (this of course is not always true, and the real question is to clarify classes of problems where resampling helps). A more pragmatic answer to the problem is to use adaptive procedures that have been proposed on the basis of heuristic considerations, where resampling is performed only when it is felt necessary, i.e. when some criterion (effective number of particles, entropy of the sample, etc.) reaches some prescribed threshold. It still remains to mathematically prove the efficiency of such adaptive procedures. The contribution of this paper is to propose an approach, based on a representation in terms of multiplicative functionals (in which importance weights are treated as particles, roughly speaking) to obtain the asymptotic variance of adaptive resampling procedures, when the sample size goes to infinity. It is then possible to see the impact of the threshold on the asymptotic variance, at least in the Gaussian case, where the resampling criterion has an explicit expressions in the large sample asymptotics.